This Phys.SE question asks why and how the speed of light is constant. I would like to ask a related, almost converse question: Given that the speed of light is constant, how could I explain to a young-earth-creationist who is still a proponent of the disgraced Barry Setterfield $c$-decay theory that the $c$-decay theory contradicts observed physical reality, as described and understood by all scientists, including those in the Physics and Astronomy fields?

Surely a variable speed of light would throw a monkey wrench into all sorts of fundamental laws or systems in physics? Surely it would be measurable on instruments today, if there was even a miniscule change in the speed of light relative to a continuous cold caesium fountain atomic clock? The fact that there is no observable change in any recent trustworthy data, is the beginning, middle, and end of the scientific argument. Am I correct?

I also wonder if $c$-decay theorists are contradicting themselves, or have sawn off the branch upon which they themselves sit; For example, I wonder this; Would we have to modify our expectations of what the doppler effect (red shift) would look like, as light comes towards us from distant stars, if the speed of light is variable? Would we not have to throw all our doppler data out, and all the distances themselves, if the speed of light is variable in some undefined way, and thus unknown and unknowable? The $c$-decay theory was spawned because redshift data indicates that the stars are too far away for their light to have reached us in the mere 6000 years the universe has existed. In short, if $c$-decay theory exists to explain distant stars to young earth creationists and help them breathe a sigh of relief, wouldn't it be slightly-less-scientifically-bogus, to just dispute the doppler effect and the distance of those stars?

2 Answers
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I'll (arguably artificially) restrict my answer to physics, avoiding the creationist side of the equation.

The question of whether physical "constants" are actually constant has been studied by physicist, and some cosmological theories predict such variations. As you say, a variation of c would change many things in physics. For example $c$ is a parameter which has an influence on the atomic spectra. The relative position some spectral lines have a relative distance proportional to the fine structure constant $\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}$, si if $c$ varies while the other parameter stays constant, one should see a change in the atomic spectra proportional to $\frac{\delta\alpha}{\alpha}\varpropto \frac{\delta c}{c}$.

This has been watched, as explained in this wikipedia article, on in this ppt presentation by Flambaum (which also look for the variation of other constants.) In short:

Spectral lines from quasar show that there might be a relative variation of $\alpha$ of the order of 10⁻⁵ since the "quasar time" (a few billion years for anyone reasonable, but probably a few thousand year for a creationist).

As you suspected, atomic clocks have also been used to monitor this effect. An atomic clock being nothing else than a very precise spectrometer. They have shown that, in the present time, $\alpha$ varies by less than 10⁻¹⁵ each year.
More recent measurement seem to reduce the possible changes of $\alpha$ even further.

I have no idea of the orders of magnitude of the variation of the c-decay "theory", but to be noticeable in historic measurements, a variation of $c$ should be clearly bigger than 10⁻² over a few centuries, which is clearly incompatible with a above two points.

Another important thing to realize is that laboratory measurements rule out a linear change in alpha over time large enough to be consistent with the Webb group's claims: Rosenband et al., 2008, 319 (5871): 1808-1812, sciencemag.org/content/319/5871/1808.abstract
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Ben CrowellJul 27 '11 at 15:55

I think that Webb is doing physics correctly but concluding erroneously. It is not a $\frac{\delta\alpha}{\alpha}$ variation but, as suggested on the page 54 of the ppt above linked , "Possible systematic effect: isotopic ratio evolution, .. "conspiracy of isotopic abundances”. This hypothesis was already mentioned in his 1998 paper. His results have the constancy of isotope ratios as an 'apriori condition'.
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Helder VelezJul 27 '11 at 16:21

At page 4 of ppt it is wrong again:"Since variation of dimensional constants cannot be distinguished from variation of units, it only makes sense to consider a variation of dimensionless constants." From dimensional analysis of $\left[G\right]=M^{-1}L^{3}T^{-2}$, $\left[\varepsilon\right]=M^{-1}Q^{2}L^{-3}T^{2}$,$\left[c\right]=LT^{-1}$. The summation of exponents of the dimension function of each field constant is zero! It means that if all the four base units concerned change by the same factor, $M=Q=L=T$ then the measuring units of field constants hold invariant. $[G]=[\varepsilon]=[c]=1$.
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Helder VelezJul 27 '11 at 16:40

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@Helder Velez: The fact that the three unitful constants you picked each happened to have exponents summing to zero is a cute coincidence, but nothing more than a coincidence. It does not hold for other dimensionful constants such as the mass of the electron, $[m_e]=M$, or Planck's constant, $[h]=ML^2T^{-1}$. I'm not sure why you refer to the three constants you quote as "field constants;" this is not standard terminology. The statement you quoted from the powerpoint is perfectly correct. For more discussion of this point, see Duff, arxiv.org/abs/hep-th/0208093 .
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Ben CrowellJul 27 '11 at 16:54

The Oklo natural nuclear reactor has also been used to check the speed of light. It was used to check the value of the fine-structure constant when the reactor was operating, about 2 billion years ago. While scientists still argue about whether it proves the constancy of c or otherwise, they do agree that any change is minute: a few parts in 100 million over 2 billion years. In other words, scientists are talking about a change of a few meters per second, compared to a (current) speed of about 300 million meters per second.

Creationists on the other hand need a massive change over a couple of thousand years (between creation and the flood). To get light from the furthest galaxies (10 billion lightyears) here in just 3000 years, you'd need c = 10^12 km/sec, rather larger than the currently accepted 300,000 km/sec! No such change has been observed anywhere.